Figures (1)  Tables (9)

    • Figure 1. 

      Transformation rules for simplifying the constraint satisfaction problem $ {\mathrm{CSP}}(\Delta) $. A pair ${\langle { {\cal{V}}},\; { {\cal{F}}} \rangle_{{i}}} $ represents the sets of constraint variables in the minimum expressions associated to the verification and the falsification, respectively, of the $ i $-th conditional $ \delta_i \in \Delta $ in the constraint $ C_i \in {\mathrm{CSP}}(\Delta) $ modeling the acceptance condition of $ \delta_i $.

    • $ \omega $ $ {\mathrm{ver}}_{\Delta_{\mathrm{ex}}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{\mathrm{ex}}}(\omega) $ $ \omega $ $ {\mathrm{ver}}_{\Delta_{\mathrm{ex}}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{\mathrm{ex}}}(\omega) $
      $ abc $ $ \{\delta_1,\delta_4\} $ $ \{\delta_3,\delta_5\} $ $ {\overline a b}c $ $ \{\delta_3\} $ $ \{\delta_4\} $
      $ ab{\overline c } $ $ \{\delta_1,\delta_5\} $ $ \{\delta_3\} $ $ {\overline a b}{\overline c } $ $ \{\delta_3\} $ $ \emptyset $
      $ a{\overline b }c $ $ \emptyset $ $ \{\delta_1,\delta_2,\delta_3\} $ $ {\overline a \overline b }c $ $ \{\delta_2,\delta_3\} $ $ \emptyset $
      $ a{\overline b }{\overline c } $ $ \emptyset $ $ \{\delta_1, \delta_2\} $ $ {\overline a \overline b }{\overline c } $ $ \{\delta_2\} $ $ \emptyset $

      Table 1. 

      Verified and falsified conditionals from $ \Delta_{\mathrm{ex}} $ (cf. Example 2).

    • $ \delta_i $ $ V_i $ $ F_i $ $ \hat{V}_i $ $ \hat{V}_i $
      $ \delta_1 = (b|a) $ $ \{ \{\delta_3\}, \{\delta_3, \delta_5\} \} $ $ \{ \{\delta_2\}, \{\delta_2, \delta_3\} \} $ $ \{ \{\delta_3\} \} $ $ \{ \{\delta_2\} \} $
      $ \delta_2 = ({\overline a }|{\overline b }) $ $ \{ \emptyset \} $ $ \{ \{\delta_1\}, \{\delta_1, \delta_3\} \} $ $ \{ \emptyset \} $ $ \{ \{\delta_1\} \} $
      $ \delta_3 = ({\overline a }|b \lor c) $ $ \{ \emptyset, \{\delta_4\} \} $ $ \{ \emptyset, \{\delta_5\}, \{\delta_1, \delta_2\} \} $ $ \{ \emptyset \} $ $ \{ \emptyset \} $
      $ \delta_4 = (a|bc) $ $ \{ \{\delta_3, \delta_5\} \} $ $ \{ \emptyset \} $ $ \{ \{\delta_3, \delta_5\} \} $ $ \{ \emptyset \} $
      $ \delta_5 = ({\overline c }|ab) $ $ \{ \{\delta_3\} \} $ $ \{ \{\delta_3\} \} $ $ \{ \emptyset \} $ $ \{ \emptyset \} $

      Table 2. 

      (Reduced) constraint inducing sets of the conditionals in $ \Delta_{\mathrm{ex}} $ (cf. Examples 2 and 3).

    • Input: (Non-empty) consistent belief base $\Delta$, constraint satisfaction problem ${\mathrm{CSP}}^+(\Delta)$
      Output: Canonical generalized tolerance partition ${\cal{Q}}^{mc}(\Delta)$, impact vector $\vec{\eta}^{mc}_\Delta$
      1 $m = 1$
      2 WHILE $\Delta \neq \emptyset$:
      3 $\eta_{\min} = 0$
      4 $\Delta' = \emptyset$
      5 FOR $\delta_i \in \Delta$:
      6 IF there is $S \in \hat{V}_i$ with $S \subseteq \bigcup_{j=1}^m \Delta_j$:
      7 $\eta_i = \min\{ \sum_{\delta_j \in S} \eta_j \mid S \in \hat{V}_i \colon S \subseteq \bigcup_{j=1}^m \Delta_j \} + 1$
      8 IF $\eta_{\min} = 0$:
      9 $\eta_{min} \leftarrow \eta_i$
      10 $\Delta' \leftarrow \Delta' \cup \{ \delta_i \}$
      11 ELSE IF $\eta_i = \eta_{\min}$:
      12 $\Delta' \leftarrow \Delta' \cup \{ \delta_i \}$
      13 ELSE IF $\eta_i \lt \eta_{\min}$:
      14 $\eta_{\min} \leftarrow \eta_i$
      15 $\Delta' \leftarrow \{ \delta_i \}$
      16 $\Delta_m \leftarrow \Delta'$
      17 $\Delta \leftarrow \Delta \setminus \Delta_m$
      18 $m \leftarrow m+1$
      19 RETURN ${\cal{Q}}^{mc}(\Delta) = (\Delta_1, \ldots, \Delta_m)$ and $\vec{\eta}^{mc}_\Delta = (\eta_1, \ldots, \eta_n)$

      Table 1. 

      Computation of the canonical generalized tolerance partition ${\cal{Q}}^{mc}(\Delta)$, and the impact vector $\vec{\eta}^{mc}_\Delta$ of a (non-empty) consistent belief base $\Delta$.

    • $ \omega $ $ {\mathrm{ver}}_{\Delta_{{\mathrm{ex}}2}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{{\mathrm{ex}}2}}(\omega) $ $ \omega $ $ {\mathrm{ver}}_{\Delta_{{\mathrm{ex}}2}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{{\mathrm{ex}}2}}(\omega) $
      $ abc $ $ \{\delta_1\} $ $ \emptyset $ $ {\overline a b}c $ $ \emptyset $ $ \{\delta_5\} $
      $ ab{\overline c } $ $ \{\delta_2\} $ $ \{\delta_1\} $ $ {\overline a b}{\overline c } $ $ \emptyset $ $ \{\delta_5\} $
      $ a{\overline b }c $ $ \{\delta_1,\delta_4\} $ $ \{\delta_2\} $ $ {\overline a \overline b }c $ $ \{\delta_5\} $ $ \{\delta_4\} $
      $ a{\overline b }{\overline c } $ $ \{\delta_3\} $ $ \{\delta_1\} $ $ {\overline a \overline b }{\overline c } $ $ \{\delta_5\} $ $ \{\delta_2,\delta_3\} $

      Table 3. 

      Verified and falsified conditionals from $ \Delta_{{\mathrm{ex}}2} $ in Example 9.

    • $ \omega $ $ {\mathrm{ver}}_{\Delta'_{{\mathrm{ex}}3}}(\omega) $ $ {\mathrm{fal}}_{\Delta'_{{\mathrm{ex}}3}}(\omega) $ $ \kappa^{mc}_{\Delta_{{\mathrm{ex}}3}}(\omega) $ $ \kappa^{mc}_{\Delta'_{{\mathrm{ex}}3}}(\omega) $
      $ abc $ $ \{\delta_1,\delta_2\} $ $ \{\delta_3\} $ 0 2
      $ ab{\overline c } $ $ \{\delta_1,\delta_3\} $ $ \{\delta_2\} $ 1 1
      $ a{\overline b }c $ $ \emptyset $ $ \{\delta_1,\delta_3\} $ 1 4
      $ a{\overline b }{\overline c } $ $ \{\delta_3\} $ $ \{\delta_1\} $ 1 2
      $ {\overline a b}c $ $ \{\delta_2\} $ $ \emptyset $ 0 0
      $ {\overline a b}{\overline c } $ $ \emptyset $ $ \{\delta_2\} $ 1 1
      $ {\overline a \overline b }c $ $ \emptyset $ $ \emptyset $ 0 0
      $ {\overline a \overline b }{\overline c } $ $ \emptyset $ $ \emptyset $ 0 0

      Table 4. 

      Minimal core c-representations of $ \Delta_{{\mathrm{ex}}3} $ and $ \Delta'_{{\mathrm{ex}}3} $ from Example 10.

    • $ {\cal{I}}^P $ $ {\cal{I}}^Z $ $ {\cal{I}}^c $ $ {\cal{I}}^{mc} $
      System P $ \checkmark$ $ \checkmark$ $ \checkmark$ $ \checkmark$
      (wSM) $ \checkmark$ $ \checkmark$ $ \checkmark$ $ \checkmark$
      (SM) $ \checkmark$ − − −
      (Rel) $ \checkmark$ $ \checkmark$ $ \checkmark$ $ \checkmark$
      (Ind) − − $ \checkmark$ $ \checkmark$
      (SynSplit) − − $ \checkmark$ $ \checkmark$
      (CRel) $ \checkmark$ $ \checkmark$ $ \checkmark$ $ \checkmark$
      (CInd) − − $ \checkmark$ $ \checkmark$
      (CSynSplit) − − $ \checkmark$ $ \checkmark$
      (CP) $ \checkmark$ $ \checkmark$ $ \checkmark$ $ \checkmark$
      (wRM) − $ \checkmark$ $ \checkmark$ $ \checkmark$
      (RM) − $ \checkmark$ − $ \checkmark$

      Table 5. 

      Inference properties which are satisfied ($ \checkmark$) or violated (−) by the inductive inference operators mentioned in this paper (System P inference $ {\cal{I}}^P $, System Z inference $ {\cal{I}}^Z $, skeptical c-inference $ {\cal{I}}^c $, and c-core closure $ {\cal{I}}^{mc} $).

    • $ \omega $ $ {\mathrm{ver}}_{\Delta_{\mathrm{bfpe}}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{\mathrm{bfpe}}} (\omega) $ $ \kappa^Z_{\Delta_{\mathrm{bfpe}}}(\omega) $ $ \kappa^{\, mc}_{\Delta_{\mathrm{bfpe}}}(\omega) $ $ \kappa^{\vec{\eta}}_{\Delta_{\mathrm{bfpe}}}(\omega) $
      $ bfpe $ $ \{ \delta_1, \delta_2, \delta_4 \} $ $ \{ \delta_3 \} $ 2 2 2
      $ bfp{\overline e } $ $ \{ \delta_1, \delta_2 \} $ $ \{ \delta_3, \delta_4 \} $ 2 3 6
      $ bf\,{\overline p }e $ $ \{ \delta_2, \delta_4 \} $ $ \emptyset $ 0 0 0
      $ bf\,{\overline p }\,{\overline e } $ $ \{ \delta_2 \} $ $ \{ \delta_4 \} $ 1 1 4
      $ b{\overline f }pe $ $ \{ \delta_1, \delta_3, \delta_4 \} $ $ \{ \delta_2 \} $ 1 1 1
      $ b{\overline f }p{\overline e } $ $ \{ \delta_1, \delta_3 \} $ $ \{ \delta_2, \delta_4 \} $ 1 2 5
      $ b{\overline f }\,{\overline p }e $ $ \{ \delta_4 \} $ $ \{ \delta_2 \} $ 1 1 1
      $ b{\overline f }\,{\overline p }\,{\overline e } $ $ \emptyset $ $ \{ \delta_2, \delta_4 \} $ 1 2 5
      $ {\overline b }fpe $ $ \emptyset $ $ \{ \delta_1, \delta_3 \} $ 2 4 4
      $ {\overline b }fp{\overline e } $ $ \emptyset $ $ \{ \delta_1, \delta_3 \} $ 2 4 4
      $ {\overline b }f\,{\overline p }e $ $ \emptyset $ $ \emptyset $ 0 0 0
      $ {\overline b }f\,{\overline p }\,{\overline e } $ $ \emptyset $ $ \emptyset $ 0 0 0
      $ {\overline b }\,{\overline f }pe $ $ \{ \delta_3 \} $ $ \{ \delta_1 \} $ 2 2 2
      $ {\overline b }\,{\overline f }p{\overline e } $ $ \{ \delta_3 \} $ $ \{ \delta_1 \} $ 2 2 2
      $ {\overline b }\,{\overline f }\,{\overline p }e $ $ \emptyset $ $ \emptyset $ 0 0 0
      $ {\overline b }\,{\overline f }\,{\overline p }\,{\overline e } $ $ \emptyset $ $ \emptyset $ 0 0 0

      Table 6. 

      Verified and falsified conditionals from $ \Delta_{\mathrm{bfpe}} $ as well as the System Z ranking model $ \kappa^Z_{\Delta_{\mathrm{bfpe}}} $, the minimal core c-representation $ \kappa^{mc}_{\Delta_{\mathrm{bfpe}}} $, and the c-representation $ \kappa^{\vec{\eta}}_{\Delta_{\mathrm{bfpe}}} $ induced by $ \vec{\eta} = (2,1,2,4) $ (cf. Example 12).

    • $ \delta_i $ $ V_i $ $ F_i $ $ \hat{V}_i $ $ \hat{F}_i $
      $ \delta_1 = (b|p) $ $ \{\{\delta_2\},\{\delta_2,\delta_4\},\{\delta_3\},\{\delta_3,\delta_4\}\} $ $ \{\emptyset,\{\delta_3\}\} $ $ \{\{\delta_2\}, \{\delta_3\}\} $ $ \{\emptyset\} $
      $ \delta_2 = (f|b) $ $ \{\emptyset,\{\delta_3\},\{\delta_3,\delta_4\},\{\delta_4\}\} $ $ \{\emptyset,\{\delta_4\}\} $ $ \{\emptyset\} $ $ \{\emptyset\} $
      $ \delta_3 = ({\overline f }|p) $ $ \{\{\delta_1\},\{\delta_2\},\{\delta_2,\delta_4\}\} $ $ \{\emptyset,\{\delta_1\},\{\delta_4\}\} $ $ \{\{\delta_1\},\{\delta_2\}\} $ $ \{\emptyset\} $
      $ \delta_4 = (e|b) $ $ \{\emptyset,\{\delta_2\},\{\delta_3\}\} $ $ \{\emptyset,\{\delta_2\},\{\delta_3\}\} $ $ \{\emptyset\} $ $ \{\emptyset\} $

      Table 7. 

      (Reduced) constraint inducing sets of the conditionals in $ \Delta_{\mathrm{bfpe}} $ (cf. Example 12).

    • $ \omega $ $ {\mathrm{ver}}_{\Delta_{{\mathrm{ex}}5}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{{\mathrm{ex}}4}}(\omega) $ $ \omega $ $ {\mathrm{ver}}_{\Delta_{{\mathrm{ex}}5}}(\omega) $ $ {\mathrm{fal}}_{\Delta_{{\mathrm{ex}}4}}(\omega) $
      $ abc $ $ \{\delta_1,\delta_2\} $ $ \emptyset $ $ {\overline a b}c $ $ \emptyset $ $ \emptyset $
      $ ab{\overline c } $ $ \emptyset $ $ \{\delta_1,\delta_2\} $ $ {\overline a b}{\overline c } $ $ \emptyset $ $ \emptyset $
      $ a{\overline b }c $ $ \{\delta_1,\delta_3\} $ $ \emptyset $ $ {\overline a \overline b }c $ $ \emptyset $ $ \emptyset $
      $ a{\overline b }{\overline c } $ $ \emptyset $ $ \{\delta_1,\delta_3\} $ $ {\overline a \overline b }{\overline c } $ $ \emptyset $ $ \emptyset $

      Table 8. 

      Verified and falsified conditionals from $ \Delta_{{\mathrm{ex}}5} $ in Example 13.